论文标题
在$ \ Mathcal {y} $ - $δ$ - 互动的$ \ mathcal {y} $上的正弦格式方程的不稳定的扭结 - 索尔顿配置文件
Unstable kink-soliton profiles for the sine-Gordon equation on a $\mathcal{Y}$-junction graph with $δ$-interaction
论文作者
论文摘要
这项工作的目的是在公制图上建立固定，扭结和扭结/反键单齿型解决方案的线性不稳定性结果，该解决方案具有由$ \ Mathcal y $ junction表示的结构。该模型考虑了$δ$ - 互动类型的图形vertex上的边界条件，换句话说，波动函数在顶点上的连续性以及用于通量的Kirchhoff型定律。结果表明，扭结和扭结/反键单齿型固定轮廓是线性（非线性）不稳定的。为此，建立了一个线性不稳定性标准，该标准在波浪周围为线性化操作员提供了足够的条件，以使其具有一对真实的正/负特征值。结果，线性稳定性分析取决于该线性操作员及其摩尔斯指数的光谱研究。对称运算符，Sturm-Liouville振荡结果和操作员分析扰动理论的扩展理论是稳定分析中的基本要素。对$ \ Mathcal e（\ Mathcal y）\ times l^2（\ Mathcal {y}）$中的正弦 - 戈登模型的本地良好性的全面研究，其中$ \ MATHCAL E（\ MATHCAL Y）\ subset Y）\ subset H^1（\ Mathcal {y}）$也是适当的能量空间。在这项研究中开发的理论具有研究公制图上其他非线性进化方程的固定波解的不稳定性的前景。
The aim of this work is to establish a linear instability result of stationary, kink and kink/anti-kink soliton profile solutions for the sine-Gordon equation on a metric graph with a structure represented by a $\mathcal Y$-junction. The model considers boundary conditions at the graph-vertex of $δ$-interaction type, or in other words, continuity of the wave functions at the vertex plus a law of Kirchhoff-type for the flux. It is shown that kink and kink/anti-kink soliton type stationary profiles are linearly (and nonlinearly) unstable. For that purpose, a linear instability criterion that provides the sufficient conditions on the linearized operator around the wave to have a pair of real positive/negative eigenvalues, is established. As a result, the linear stability analysis depends upon of the spectral study of this linear operator and of its Morse index. The extension theory of symmetric operators, Sturm-Liouville oscillation results and analytic perturbation theory of operators are fundamental ingredients in the stability analysis. A comprehensive study of the local well-posedness of the sine-Gordon model in $\mathcal E(\mathcal Y) \times L^2(\mathcal{Y})$ where $\mathcal E(\mathcal Y) \subset H^1(\mathcal{Y})$ is an appropriate energy space, is also established. The theory developed in this investigation has prospects for the study of the instability of stationary wave solutions of other nonlinear evolution equations on metric graphs.
